%I #9 Feb 24 2022 09:58:12

%S 1,0,1,0,0,0,2,0,1,0,1,0,4,0,3,0,3,1,7,0,6,1,6,1,11,0,11,2,11,3,19,1,

%T 18,3,18,5,30,4,28,6,30,10,45,6,40,11,46,16,63,11,60,19,69,25,88,18,

%U 86,32,97,36,121,32,123,47,131,55,164,49,164,69,181,80

%N Number of partitions of n into 8 distinct primes (counting 1 as a prime).

%p b:= proc(n, i) option remember; series(`if`(n=0, 1,

%p `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i-1)))(

%p `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 9)

%p end:

%p a:= n-> coeff(b(n, numtheory[pi](n)), x, 8):

%p seq(a(n), n=59..130); # _Alois P. Heinz_, Feb 24 2021

%t b[n_, i_] := b[n, i] = Series[If[n == 0, 1,

%t If[i < 0, 0, Function[p, If[p > n, 0, x*b[n - p, i - 1]]][

%t If[i == 0, 1, Prime[i]]] + b[n, i - 1]]], {x, 0, 9}];

%t a[n_] := Coefficient[b[n, PrimePi[n]], x, 8];

%t Table[a[n], {n, 59, 130}] (* _Jean-François Alcover_, Feb 24 2022, after _Alois P. Heinz_ *)

%Y Cf. A008578, A036497, A219202, A341951, A341973, A341974, A341975, A341976, A341977, A341978, A341980, A341981.

%K nonn

%O 59,7

%A _Ilya Gutkovskiy_, Feb 24 2021